Abstract

We study here different fractional versions of the compound Poisson process. The fractionality is introduced in the counting process representing the number of jumps as well as in the density of the jumps themselves. The corresponding distributions are obtained explicitly and
proved to be solution of fractional equations of order less than one. Only in the final case treated
in this paper, where the number of jumps is given by the fractional-difference Poisson process
defined in Orsingher and Polito (2012), we have a fractional driving equation, with respect to the time argument, with
order greater than one. Moreover, in this case, the compound Poisson process is Markovian and
this is also true for the corresponding limiting process. All the processes considered here are proved to be compositions of continuous time random
walks with stable processes (or inverse stable subordinators). These subordinating relationships
hold, not only in the limit, but also in the finite domain. In some cases the densities satisfy
master equations which are the fractional analogues of the well-known Kolmogorov one.

1. Introduction and Preliminary Results

The fractional Poisson process (FPP), which we will denote by , , , has been introduced in [1], by replacing, in the differential equation governing the Poisson process, the time derivative with a fractional one. Later, in [2, 3], it was proved to be a renewal process with Mittag-Leffler distributed waiting times (and therefore with infinite mean). In [4] it has been expressed as the composition of a standard Poisson process with the fractional diffusion , independent of . A full characterization of in terms of its finite multidimensional distributions can be found in [5]. In [6] the coincidence between and the fractal time Poisson process (FTPP) defined as has been proved, where , is the inverse of the stable subordinator of index (with parameters , , , in the notation of [7], that we will adopt hereafter). Thus, the process is characterized by the following Laplace pairs:
where is the Mittag-Leffler function of parameters , and is the density of . The inverse stable subordinator is defined by the following relation:
and therefore we get
where is the density of .

We will make use also of different forms of FPP such as the alternative fractional Poisson process in [8] and the fractional-difference Poisson process in [9].

In this paper we study several fractional compound Poisson processes and, to help the reader, we list the acronyms used throughout the paper by the end of the paper.

The first form of fractional compound Poisson process has been introduced in [10], in the form of a continuous time random walk with infinite-mean waiting times (see also [11]). This corresponds to the following random walk time changed via the FTPP, that is,
with , are i.i.d. random variables, independent from and . The last assumption (that we will adopt throughout the paper) corresponds to the so-called uncoupled case.

In [6] it is proved that subordinating random walk to the fractional Poisson process , , produces the same one-dimensional distribution. The (generalized) density function of can be expressed as
where the first term refers to the probability mass concentrated in the origin, denotes the Dirac delta function, and denotes the density of the absolutely continuous component. The function given in (1.5) satisfies the following fractional master equation, that is,
where is the Caputo fractional derivative of order (see, for example, [12]) and the random variables , have continuous density .

We also recall the following result proved in [13] for the rescaled version of the time-fractional compound Poisson process (hereafter TFCPP): if the random variables , are centered and have finite variance, then
where is a standard Brownian motion and denotes weak convergence.

A detailed exposition of the theory of TFCPP and continuous time random walks can be found in [14, 15], where the density is expressed in terms of successive derivatives of the Mittag-Leffler function as follows:
where is the th convolution of the density of the r.v.'s .

A further asymptotic result has been proved in [15], under the assumption that the density of the jump variables (which we will denote, in this special case, as ) behaves asymptotically as
where denotes the Fourier transform. In this case the TFCPP is defined as and the rescaled version displays the following weak convergence:
for , s.t. . The characteristic function of the limiting process is given by
and thus it can be represented as , where is a symmetric -stable process with parameters , , . For , the inverse stable subordinator is not Markovian as well as not Lévy (see [10]) and the same is true for , as remarked in [15]; moreover, the density of the latter is the solution to the space-time fractional equation:
where denotes the Riesz-Feller derivative of order (see [16]). Thus, in the special case , it reduces to the composition of a Cauchy process with .

Finally, we recall the following result proved in [17]: under the assumption of heavy tailed r.v.’s representing the jumps, that is,
the following convergence holds, as ,
In (1.14) is a -stable Lévy process with density and characteristic function
under the assumption that . The density of the limiting process is proved to satisfy the following time and space fractional equation:
where the fractional derivatives are intended in the Riemann-Liouville sense (see [12], formulae (2.2.3) and (2.2.4), page 80).

We present, in this paper, different versions of the compound Poisson process (CPP), fractional (under different acceptions) with respect to time and space; we provide for them analytic expressions of the distributions and some composition relationships with stable and inverse-stable processes, holding not only in the scaling limit, but also in the finite domain.

Tables 1 and 2 provide a summary of these results in the finite and asymptotic domains, respectively.

Table 1: Main results in finite domain.

Table 2: Main results in asymptotic domain.

We assume here exponential jumps (generalized later to Mittag-Leffler), since this allows to obtain explicit equations (fractional in most cases) driving these fractional CPP's for any finite value of the time and space arguments. This kind of explicit formulae, together with the knowledge of the related governing differential equations, is of great importance in many actuarial applications (see, for example, [18], Section 4.2). In risk theory it is related to the Tweedie's compound Poisson model (see [19]). The hypothesis of exponential jumps has been widely applied also in other fields: in natural sciences it leads to the so-called compound Poisson-Gamma model, which is used for rainfall prediction (see, for example, [20]).

2. Time-Fractional Compound Poisson Processes

We consider different forms of TFCPP, starting with the more familiar one given in (1.4) and then comparing the results with those obtained for an alternative definition of FPP.

2.1. The Standard Case

In order to get a form of the density of the TFCPP more explicit than (1.8), we assume that the 's are exponentially distributed: in this case it can be expressed in terms of the generalized Mittag-Leffler function:
where is the rising factorial (or Pochhammer symbol). Moreover, we can obtain the fractional partial-differential equation satisfied by the density of its absolutely continuous component.

Theorem 2.1. The process
with , , independent and exponentially distributed with parameter , has the following distribution:
where
The function given in (2.4) satisfies the following partial differential equation:
where denotes the Caputo fractional derivative with the conditions

Proof. Formula (1.8) can be rewritten by considering that and using the expression of in terms of generalized Mittag-Leffler functions (see [21]), that is,
In order to derive (2.5), we evaluate the following partial derivatives of (2.4):
By inserting (2.8) in (2.5), the equation is satisfied. Finally, it can be easily verified that the initial condition holds. In order to check the second condition in (2.6), we integrate with respect to :
where, in the last step, we have applied formula (2.30) of [21], for .

2.1.1. The Nonfractional Case

From (2.4), we obtain the distribution of the standard CPP, defined as , under the assumption of exponential jumps , which reads
where
is the Wright function. Equation in [18] provides another expression of in terms of the modified Bessel function. The density (2.11) satisfies the following equation:
with conditions
as can be easily verified directly.

Now we recall the following subordination law presented in [6] in a more general setting:
where , is the inverse stable subordinator defined by (1.2). We give an explicit proof of (2.15), which will be useful to prove analogous results in the next sections. We start with the evaluation of the Laplace transform (hereafter denoted by ) of with respect to : by considering the probability generating function of , that is,
we get
Formula (2.17), Laplace transformed with respect to , gives
which can be rewritten as
where . Thus, by inverting the double Laplace transform, we get
Now it is also easy to derive (2.5), since we can write in particular from (2.20) that
and thus we get
Indeed, it is well known that is governed by the following equation:
By integrating by parts and applying the initial condition, (2.22) becomes

2.2. An Alternative Case

We consider now a different model of TFCPP, based on the alternative definition of FPP given in [4], that is,
The process with the above state probabilities plays a crucial role in the evolution of some random motions (see [22]) and can be considered as a fractional version of the Poisson process because its probability generating function (displayed below) satisfies a fractional equation (see formula (4.5) of [4]). The distribution (2.25) can be interpreted as a weighted Poisson distribution (for the general concept of discrete weighted distribution see, e.g., [23], page 90, and the references cited therein) and, as explained in [8], the weights that do not depend on ; actually we have
where , (for all ) and , are the distribution of the standard Poisson process with intensity . We also recall [24] where one can find a sample path version of the weighted Poisson process.

We remark that the corresponding process is not Markovian, as , and moreover is not a renewal. Nevertheless, it is, for some aspects, more similar to the standard Poisson process than . For example, the rate of the asymptotic behavior of its moments is the same as for .

The moment generating function is given by
so that we get
By applying the following asymptotic formula of the Mittag-Leffler function
(see, for example, [25] or [26]) we get
while for the mean value behaves asymptotically as .

We define the alternative TFCPP as
where again ’s are i.i.d. with exponential distribution, independent from . Under this assumption we obtain the following result on the distribution of .

Theorem 2.2. The process defined in (2.31), with , , independent and exponentially distributed with parameter , has the following distribution:
where

Proof. The density (2.33) can be obtained as follows:
Moreover, one can check that
and this completes the proof.

Remark 2.3. For , formula (2.33) reduces to (2.11). We note that, as happens for the standard case, the density in (2.33) is expressed in terms of a single Wright function instead of an infinite sum of generalized Mittag-Leffler functions (as for the process ). Nevertheless, the presence of a Mittag-Leffler in the denominator does not allow to evaluate the equation satisfied by .

2.2.1. Asymptotic Results

The analogy with the standard case is even more evident in the asymptotic behavior of the rescaled version of (2.31). Under the assumption (1.9) for the r.v.'s , we can prove that, as , s.t. (not depending on ),
where is a symmetric -stable Lévy process with and . Indeed, the characteristic function of (2.36) can be written as
By considering the generalized binomial theorem, we get from (2.37) that
Therefore, the limiting process is represented by the -stable process with characteristic function , instead of the subordinated process obtained in the limit when considering the FPP ; note that coincides with in (1.10). It is clear that the dependence on is limited to the scale parameter; the space-fractional equation satisfied by its density is therefore given by
instead of (1.12). For the density of the limiting process reduces to a Cauchy with scale parameter .

3. Space-Fractional Compound Poisson Process

We define now a space-fractional version of the compound Poisson process (which we will indicate hereafter by SFCPP): indeed, its distribution satisfies (2.5), but with integer time derivative and fractional space derivative. We consider the standard CPP
where, as usual, is a standard Poisson process with parameter and the random variables have the following heavy tail distribution:
for . The Laplace transform of (3.2) is . The distribution of given in (3.2) is usually called Mittag-Leffler and coincides with the geometric-stable law of index (hereafter ) with parameters and (see [27]). The density of is given by
with Laplace transform
Note that (3.3) coincides with the density of the th event waiting time for the fractional Poisson process (see [21]). It is easy to check that the variable displays the asymptotic behavior (1.13).

Theorem 3.1. The process defined in (3.1), with , , independent and distributed according to (3.2), has the following distribution:
where
The density (3.6) satisfies the following equation:
with conditions
The following composition rule holds for the one-dimensional distribution of (3.1):
where is the stable subordinator defined in (1.1) and is the standard CPP.

Proof. We start by noting that the absolutely continuous part of the distribution is defined in , with the exclusion of , where only the discrete component gives some contribution.In order to check (3.7) we evaluate the following fractional derivatives, arguing as in the proof of Theorem 2.1:
The initial condition is immediately satisfied by (3.6), while the second condition in (3.8) can be verified as follows:
which, for , becomes . The composition rule given in (3.9) can be verified by taking the Laplace transform of ,
which Laplace transformed with respect to gets
Thus,
so that, by (1.1), we get
and formula (3.9) follows.

Remark 3.2. Equation (3.15) yields an alternative proof of (3.7) noting that the density of satisfies the following equation (where the space-fractional derivative is defined now in the Caputo sense):
Indeed, we get
By considering (3.9) together with (1.2), we can write the following relationship:
while for the first version of TFCPP we had, from (2.15), that .We finally note that the process is still a Markovian and Lévy process, since it is substantially a special case of CPP.

3.1. Special Cases

For , since the 's reduce to exponential r.v.'s, from (3.6) and (3.7) we retrieve the results (2.11) and (2.13) valid for the standard CPP, under the exponential assumption for 's. As a direct check of (3.9), we can consider the special case , so that the law can be written explicitly as the density of the first passage time of a standard Brownian motion through the level . Then by considering (3.15) we can write
where the last equality holds by (2.11)-( 2.12) in [28]; then, by in [12], we get

3.2. Asymptotic Results

We study now the asymptotic behavior of the rescaled version of defined as
for . The Fourier transform of the r.v.'s, , for any , is given by
(see [27], formula (2.4.1)), which, in the limit, behaves as
where . Thus, the characteristic function of (3.21) can be written as
for , s.t. We can conclude that
where the limiting process is represented, in this case, by an -stable subordinator with parameters , , , whose density satisfies

4. Compound Poisson Processes Fractional in Time and Space

We consider now together the results obtained in the previous sections, by defining a CPP fractional both in space and time (STFCPP), that is,
where 's are i.i.d. with density (3.2) and , is again the FPP.

Theorem 4.1. The process , , defined in (4.1) has the following distribution:
where
The density solves the following equation:
with conditions
The following equality of the one-dimensional distributions holds:

Proof. In order to check (4.4) we evaluate the following fractional derivatives:
By some algebraic manipulations we finally get (4.4). While the initial condition is trivially satisfied, the second condition in (4.5) can be checked as follows:
which, for , becomes .The relationship (4.6) can be checked by evaluating the double Laplace transform of as follows:
We then rewrite formula (4.9) as
and we follow the same lines which lead to (2.15) to get the conclusion.

Remark 4.2. For formulae (4.3) and (4.4) coincide with (2.4) and (2.5), while for we get (3.6) and (3.7).From (4.6), by considering (1.2), we get the following relation:
where is the inverse stable subordinator.

4.1. Asymptotic Results

For the rescaled version of we obtain the following asymptotic result, which agrees with and proved in [17]: the characteristic function of the process
can be written as
By applying formula (3.23) we conclude that (4.13) converges, for s.t. , to
so that the process converges weakly to the -stable subordinator , composed with the inverse -stable subordinator . Indeed, the characteristic function of can be evaluated as follows:
where is the law of and . By inverting the Laplace transform in (4.15) we get (4.14). The density of satisfies the following equation:
as can be easily seen from (4.15) (see also [29]). A relevant special case of this result can be obtained by taking , so that the composition is proved to display a Lamperti-type law (see on this topic [30, 31]); therefore, the latter can be seen as the weak limit of the STFCPP.

We note that in the particular case , the Fourier transform (4.14) reduces to (3.24) and correspondingly (4.16) coincides with (3.26).

Finally, we consider the case where we have in place of . If the jumps are Mittag-Leffler distributed, we get the following space-time fractional CPP:
whose distribution is given by
where
The rescaled version of (4.17) is defined as
for , s.t. , where again denotes the -stable subordinator with characteristic function given in (3.24) (last line). Thus, in the limit, the fractional nature of the counting process does not exert any influence, in analogy with the result given in (3.25).

5. Fractional-Difference Compound Poisson Process

We present now a final version of the fractional CPP, where the fractionality of the counting process is referred to the difference operator involved in the recursive equation governing its distribution. Let denote the standard backward shift operator, , and let be a fractional parameter in , then the fractional recursive differential equation
has been introduced in [9]. In (5.1) the following definition of the fractional difference operator of a function has been used (see [12], formula (2.8.2), page121):
where is the falling factorial. We use the notation , and we have
It can be proved that is not a renewal process, by verifying that the density of the th event waiting time cannot be expressed as th convolution of i.i.d. random variables. Nevertheless, is a Lévy process, with infinite expected value for any . Moreover, by (5.3), one can check that (as )
instead of for , as for the standard or the time-fractional Poisson process. We can obtain (5.1) from (5.4) by taking into account that the increments are independent and stationary.

Let us define the corresponding fractional-difference compound Poisson process (hereafter FCPP) as
so that we can obtain, under the assumption of i.i.d. exponential 's, the distribution of together with the differential equation which is satisfied by its absolutely continuous component.

Theorem 5.1. For , the distribution of the process defined in (5.5), with , , independent and exponentially distributed with parameter , is given by
where